partition_ge_ground
plain-language theorem explainer
The partition function Z for any system of energy levels at positive inverse temperature β is strictly positive. Researchers deriving statistical mechanics from J-cost functionals would cite this to guarantee that the Boltzmann distribution is normalizable. The proof is a one-line wrapper applying the positivity lemma for the partition function.
Claim. For any system $sys$ consisting of a non-empty list of energy levels and any inverse temperature parameter $β > 0$, the partition function $Z(sys, β) > 0$.
background
In Recognition Science the Boltzmann distribution is obtained by cost-weighted selection under a fixed total J-cost constraint. A System is a structure holding a non-empty list of EnergyLevel objects. The partition function is the sum of Boltzmann factors exp(-β E_i) over those levels, with β acting as the Lagrange multiplier that enforces the cost constraint (see MODULE_DOC on J-cost derivatives and temperature). This positivity result rests on upstream cost non-negativity from ObserverForcing.cost and MultiplicativeRecognizerL4.cost, together with the ledger factorization in DAlembert.LedgerFactorization.of.
proof idea
This is a one-line wrapper that applies the lemma partition_positive sys beta hb.
why it matters
The result closes the basic positivity check needed for the partition function to be well-defined, ensuring the free energy and normalized probabilities exist in the J-cost derivation of statistical mechanics. It directly supports the module's target of obtaining P_i = exp(-β E_i)/Z from recognition cost maximization. The declaration sits inside the THERMO-001 derivation chain that links J-cost to the emergence of temperature; no downstream uses are recorded yet.
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